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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2025 • Question 6

Question

Math 6210 • Real Analysis Spring 2025 • Question 6

Let \(\mathcal {B}\) be the Borel \(\sigma \)-algebra on \(\mathbb {R}\) and \(\mathcal {M}\) be a \(\sigma \)-algebra on \(\mathbb {R}\). Prove that \(f:(\mathbb {R},\mathcal {M})\to (\mathbb {R},\mathcal {B})\) is measurable for all monotone functions \(f\) iff \(\mathcal {M}\) contains the Borel \(\sigma \)-algebra.

Raw LaTeX

Let \(\mathcal{B}\) be the Borel \(\sigma\)-algebra on \(\mathbb{R}\) and \(\mathcal{M}\) be a \(\sigma\)-algebra on \(\mathbb{R}\). Prove
that \(f:(\mathbb{R},\mathcal{M})\to(\mathbb{R},\mathcal{B})\) is measurable for all monotone functions \(f\) iff
\(\mathcal{M}\) contains the Borel \(\sigma\)-algebra.

Topic Tags

Borel Sets and Sigma-Algebras Measurable Functions